### Abstract

In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International

Original language | English (US) |
---|---|

Pages (from-to) | 3529-3549 |

Number of pages | 21 |

Journal | International Journal of Innovative Computing, Information and Control |

Volume | 6 |

Issue number | 8 |

State | Published - Aug 2010 |

### Fingerprint

### Keywords

- Bi-level programming
- Gradient approximation method
- Nelder-Mead algorithm
- Optimum toll setting problem
- Penalty function methods
- Quasi-Newton-type method

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Information Systems
- Software
- Theoretical Computer Science

### Cite this

*International Journal of Innovative Computing, Information and Control*,

*6*(8), 3529-3549.

**Comparison of algorithms for solving a bi-level toll setting problem.** / Kalashnikov, Vyacheslav; Camacho, Fernando; Askin, Ronald; Kalashnykova, Nataliya.

Research output: Contribution to journal › Article

*International Journal of Innovative Computing, Information and Control*, vol. 6, no. 8, pp. 3529-3549.

}

TY - JOUR

T1 - Comparison of algorithms for solving a bi-level toll setting problem

AU - Kalashnikov, Vyacheslav

AU - Camacho, Fernando

AU - Askin, Ronald

AU - Kalashnykova, Nataliya

PY - 2010/8

Y1 - 2010/8

N2 - In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International

AB - In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International

KW - Bi-level programming

KW - Gradient approximation method

KW - Nelder-Mead algorithm

KW - Optimum toll setting problem

KW - Penalty function methods

KW - Quasi-Newton-type method

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UR - http://www.scopus.com/inward/citedby.url?scp=77956237075&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77956237075

VL - 6

SP - 3529

EP - 3549

JO - International Journal of Innovative Computing, Information and Control

JF - International Journal of Innovative Computing, Information and Control

SN - 1349-4198

IS - 8

ER -